论文标题
分区中的挂钩长度和符号内容
Hook length and symplectic content in partitions
论文作者
论文摘要
$ gl（n，\ mathbb {c}）$，$ sp（2n）$或$ so（n）$的不可约表示的尺寸由相应分区的相应挂钩长度和内容公式给出。第一作者受到Nekrasov-Okounkov公式的启发，对涉及挂钩长度和符号/正交内容的类似表达式的组合解释进行了猜想。我们证明了这些猜想的特殊情况。在此过程中，我们表明，与所有符号内容的$ n $的分区非零具有平等性，分区$ n $均为不同的零件。我们还向贝克型同伴展示了这种身份。在这种情况下，我们将分区数量的奇偶校验分为不同（分别（甚至）等级的不同部分。我们研究钩长的总和与分区二进制表示中的反转之和之间的联系。此外，我们介绍了一个新的分区统计量，即分区的$ x $ ray列表，并探索了其与给定楼梯分区中最大包含的不同分区以及分区的联系。
The dimension of an irreducible representation of $GL(n,\mathbb{C})$, $Sp(2n)$, or $SO(n)$ is given by the respective hook-length and content formulas for the corresponding partition. The first author, inspired by the Nekrasov-Okounkov formula, conjectured combinatorial interpretations of analogous expressions involving hook-lengths and symplectic/orthogonal contents. We prove special cases of these conjectures. In the process, we show that partitions of $n$ with all symplectic contents non-zero are equinumerous with partitions of $n$ into distinct even parts. We also present Beck-type companions to this identity. In this context, we give the parity of the number of partitions into distinct parts with odd (respectively, even) rank. We study the connection between the sum of hook-lengths and the sum of inversions in the binary representation of a partition. In addition, we introduce a new partition statistic, the $x$-ray list of a partition, and explore its connection with distinct partitions as well as partitions maximally contained in a given staircase partition.
